臺灣博碩士論文加值系統

長度偏差取樣 (Length biased sampling) 是一個在各領域上被廣泛使用的取樣方法, 例如: 經濟、 工業可靠度、 應用流行病學與癌症預防。 因為長度偏差取樣所得之資料結構獨特, 不同於傳統倖存分析資料, 所以無法透過以往的倖存方法直接分析。 本文在右設限 (right censoring) 機制下針對此取樣資料作統計分析, 並透過半母數模型 (semi-parametric model) 探討因子與倖存時間 (survival time) 的關係。 半母數模型相對於母數模型 (parametric model) 的最大優勢在於擁有較平穩的估計, 且不需擔心因不當的母體假設而伴隨的統計誤判, 但在大部分的情況卻需要處理無窮維度的干擾變數。 另一方面, 根據模型在統計推論之便利性與對資料配適之合理性, 文獻中最常被運用於倖存分析的半母模型為比例風險模型 (proportional hazard model) 與比例勝算模型(proportional odds model)。 但不同的模型使用上需要不同的估計方法而造成使用者在實務操作上的不便。 綜合上述, 本文所選的半母數模型為轉換模型 (transformation model), 乃近年來被學者們廣泛討論的半母 數方法之一。 轉換模型為一個廣義的線性模型其中包含比例風險模型與比例勝算模型, 故能更彈性地分析資料。 本文針對轉換模型, 在此提出統一的估計法方且適用於轉換模型中包含的所有特例, 並利用無母數之最大概似估計量 (nonparametric maximum likelihood estimator; NPMLE ) 做參數推估, 再者, 當固定迴歸參數時干擾參數真備 self-consistency 估計值之特性。 此外, 文中提供演算法操作流程藉以實施參數與變異數之估計, 並證明 NPMLE 對應概似函數存在且隨樣本數遞增將一致收斂至所感興趣母體之真實參數以及近似服從於 tight Gaussian process。 文中設計模擬藉以探討右設限率以及不同模型間的變化對估計量所造成的影響, 且透過模擬實務範例與現實議題連結。 觀看 NPMLE 與文獻方法比較之結果, 顯示 出在各設定下 NPMLE 皆可提供優越的統計性質並隨樣本數遞增效益更佳彰顯。在真實資料分析中則採用阿茲罕默症資料與 Channing House 資料, 並以所提與文獻中現成方法做分析並探討其結果。

Length biased sampling has been widely used in various fields, such as epidemiology, cancer prevention trials and industrial reliability. Because of the sampling design, structure of length biased data is different to traditional survival data and the method of traditional survival analysis cannot be directly applied. Thus, we propose an estimation of semi-parametric transformation model under the right censored length biased data. The reasons to analyzed survival data based on semiparametric model is its flexibility. Besides, according to the convenience of statistical inference and reasonableness of model fitting, proportional hazards model and proportional odds models are the most commonly applied in the literature of survival analysis. However, in most situations, we need to deal with the infinite dimensional nuisance parameters. In addition, different models requires a different estimation methods, which leads inconvenience to the user on the practical. Hence, we focus on semi-parametric transformation model and propose a unified estimation procedure. Semi-parametric transformation model is a flexible model which contains proportional hazard model and proportional odd model has been widely discussed by scholars in recent years. The estimation procedure applies nonparametric maximum likelihood estimator (NPMLE) under the full likelihood in order to improve the efficiency. Besides, when the regression parameters is fixed, nuisance parameters is estimated by the self-consistency estimating equation, and we provide algorithm for implement estimation procedures. On the theoretical perspectives, we prove NPMLE provide existence, consistency property and converges to a tight Gaussian process. In simulation, we investigate the performance under different model setting, censoring rate and sample size. Compare NPMLE with methods in literature, the simulation results show that the proposed estimator provide superior efficiency. And this fact becomes more evident as sample size increasing. In the real data analysis, Alzheimer’s disease data and Channing House data are analyzed and the results of analysis has been included in the text.

第一章 介紹 1
第二章 估計流程、 大樣本性質、 演算法 4
2.1 在轉換模型下之最大概似估計量 4
2.2 估計方法 6
2.3 大樣本性質 9
2.3.1 最大概似估計量之存在性及一致收斂性 11
2.3.2 近似分配 12
2.4 演算法 15
2.4.1 計算參數 15
2.4.2 計算樣本標準誤 16
第三章 模擬 17
第四章 真實資料分析 25
4.1 阿茲海默症資料 25
4.2 Channing House 資料 27
第五章 結論及探討 30
附錄 相關證明流程 32
A.1 最大概似估計量之存在性及一致收斂性 32
A.2 近似分配 46

Addona, V. and Wolfson, D. B. (2006). A formal test for the stationarity of the
incidence rate using data from a prevalent cohort study with follow-up. Lifetime
Data Analysis, 12(3):267–284.
Asgharian, M., M’Lan, C. E., andWolfson, D. B. (2002). Length-biased sampling
with right censoring: An unconditional approach. Journal of the American
Statistical Association, 97(457):201–209.
Asgharian, M. and Wolfson, D. B. (2005). Asymptotic behavior of the unconditional
npmle of the length-biased survivor function from right censored prevalent
cohort data. The Annals of Statistics, 33(5):2109–2131.
Bennett, S. (1983a). Analysis of survival data by the proportional odds model.
Statistics in Medicine, 2(2):273–277.
Bennett, S. (1983b). Log-logistic regression models for survival data. Applied
Statistics, 32(2):165–171.
Chen, H. Y. and Little, R. J. (1999). Proportional hazards regression with missing
covariates. Journal of the American Statistical Association, 94(447):896–908.
Chen, K., Jin, Z., and Ying, Z. (2002). Semiparametric analysis of transformation
models with censored data. Biometrika, 89(3):659–668.
Chen, Y. H. (2009). Weighted breslow-type and maximum likelihood estimation
in semiparametric transformation models. Biometrika, 96(3):591–600.
Chen, Y. H. and Zucker, D. M. (2009). Case-cohort analysis with semiparametric
transformation models. Journal of Statistical Planning and Inference,
139(10):3706–3717.
Cheng, S. C., Wei, L. J., and Ying, Z. (1995). Analysis of transformation models
with censored data. Biometrika, 82(4):835–845.
Cheng, Y. J. and Huang, C. Y. (2014). Combined estimating equation approaches
for semiparametric transformation models with length-biased survival data.
Biometrics.
Cox, D. R. (1972). Regression models and life tables. Journal of the Royal
Statistical Society: Series B (Statistical Methodology), 34(2):187–220.
De UÑA-ÁLvarez, J., Otero-GirÁLdez, M. S., and Alvarez-Llorente, G. (2003).
Estimation under length-bias and right-censoring: an application to unem-ployment duration analysis for married women. Journal of Applied Statistics,
30(3):283–291.
Gail, M. H. and Benichou, J. (2000). Encyclopedia of epidemiologic methods.
John Wiley &; Sons Ltd.
Huang, C. Y. and Qin, J. (2011). Nonparametric estimation for length-biased and
right-censored data. Biometrika, 98(1):177–186.
Kalbfleisch, J. D. and Prentice, R. L. (1973). Marginal likelihoods based on cox’s
regression and life model. Biometrika, 60(2):267–278.
Keiding, N. (1991). Age-specific incidence and prevalence: a statistical perspective.
Journal of the Royal Statistical Society: Series A (Statistics in Society),
154(3):371–396.
Kvam, P. (2008). Length bias in the measurements of carbon nanotubes. Technometrics,
50(4):462